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Question:

If $f(x)=\left\{\begin{array}{cl}\frac{\sin ^{-1} x}{x}, & x \neq 0 \\ k & , x=0\end{array}\right.$ is continuous at $x=0$, write the value of $k$.

Solution:

Given, $f(x)=\left\{\begin{array}{l}\frac{\sin ^{-1} x}{x}, x \neq 0 \\ k, x=0\end{array}\right.$

If $f(x)$ is continuous at $x=0$, then

$\lim _{x \rightarrow 0} f(x)=f(0)$

$\Rightarrow \lim _{x \rightarrow 0}\left(\frac{\sin ^{-1} x}{x}\right)=f(0)$

$\Rightarrow \lim _{x \rightarrow 0}\left(\frac{\sin ^{-1} x}{x}\right)=k$

$\Rightarrow k=1 \quad\left[\because \lim _{x \rightarrow 0}\left(\frac{\sin ^{-1} x}{x}\right)=1\right]$